It is known that in a Hilbert space given a compact set the closure of its convex hull is compact This is true for any locally convex topological vector space, see e.g. [1]}.. In finite dimensional Euclidean spaces even a stronger result holds that the convex hull itself of a compact set is compact, a conclusion that follows immediately from Carathéodory's Theorem. Here we investigate compact sets and the closure of their convex hulls in the setting of a complete \(\operatorname{CAT}(0)\)  space. \(\operatorname{CAT}(0)\)  spaces are geodesically connected metric spaces where every geodesic triangle is at least as thin as its comparison triangle in the Euclidean plane. This property
generalizes in a way the notion of nonpositive sectional curvature for a Riemannian manifold. Such a definition for curvature in a metric space is due to Alexandrov [2]} and it was later popularized by Gromov [3]}, [4]}. Often one refers to a \(\operatorname{CAT}(0)\)  space as a metric space of nonpositive curvature in the sense of Alexandrov (see [5]}, [6]}). A Hilbert space is a particular example of a complete \(\operatorname{CAT}(0)\)  space.
